Using the Power Law Model for Data Analysis in RGA

[Editor's Note: This article has been updated
since its original publication to reflect a more recent
version of the software interface.]

The Power Law model is a popular method for analyzing the reliability of
complex repairable systems in the field. In this article we first give a brief
introduction to the Power Law model and we then give an example
that shows how to use Power Law model
in RGA to estimate the conditional
reliability of a group of systems.

Introduction to the Power Law

In the real world, some systems consist of many components. A failure of one
critical component would bring down the whole system. After the component is
repaired, the system has been repaired. However, because there are many other
components still operating with various ages, the system is not put back into a
like new condition after the repair of the single component. The repair of a
single component is only enough to get the system operational again, which means
the system reliability is almost the same as that before it failed. For example, a
car is not as good as new after the replacement of a failed water pump. This kind
of repair is called minimal repair. Distribution theory does not
apply to the failures of a complex system; the intervals between failures of a
complex repairable system do not follow the same distribution. Rather, the sequence
of failures at the system levels follows a non-homogeneous Poisson process
(NHPP).

When the system is first put into service, its age is 0. Under the NHPP, the
first failure is governed by a
distribution F(x)
with failure rate r(x). Each succeeding
failure is
governed by the intensity function u(x) of
the process. Let
t be the age of the system and assume
that Δt is very small. The probability that a
system of age t fails between
t
and t + Δt is given by the intensity
function u(t)Δt; the failure
intensity u(t) for the NHPP has the same
functional form as the failure rate governing the first system failure. If the
first system failure follows the Weibull distribution, the failure rate is:

and the system intensity function is:

This is the Power Law model. The Weibull distribution governs the first system
failure and the Power Law model governs each succeeding system failure. The Power
Law mean value function is:

Here T is the system operation end
time and N(T) is the number of failures
over time 0 to T.

Parameter Estimation

For the Power Law model, there are two parameters
λ
and β. The MLE estimation for them is given by:

where:

K is the number of systems under
study.

q is the index of
the qth system under
observation from time Sq
to Tq.

Nq is the number of failures
experienced by the qth system.

Xi,q is the age of this system
at the ith occurrence of
failure.

0ln0 is defined to be 0.

If XNi,q = Tq
then the data on the qth system is said
to be failure terminated and Tq
is a random variable with Nq fixed. If
XNi,q < Tq
then the data on the qth system is said
to be time terminated and Nq is
a random variable.

In general, these equations cannot be solved explicitly
for
and ,
but must be solved by iterative procedures.
If S1 = S2 = ... = S
q = 0 and
T1 = T2 = ... =
Tq(q = 1,2,...,K), then the MLE
for
and
are in closed form:

Conditional Reliability

By using the Power Law model, it is easy to estimate the probability that the
system will survive to age t + d without failure
given that the current system age is t. That is,
the equation to get the mission reliability for a system of
age t and mission
time d is:

Confidence Bounds for Reliability

There are two kinds of reliability confidence bounds available
in RGA: the Fisher Matrix confidence bounds and the Crow confidence
bounds.

Fisher Matrix Confidence Bounds

The Fisher Matrix confidence bounds on reliability are given by:

where:

The variance can be calculated using the Fisher Information Matrix
where Λ is the natural log-likelihood function.

Crow Confidence Bounds

The Crow confidence bounds on reliability with failure terminated data are given
by:

where:

The values of ρ1
and ρ2 can be obtained by finding the
solution c
to
for
and , respectively.

where:

The Crow confidence bounds on reliability with time terminated data are given
by:

where:

The values of Π1
and Π2 can be obtained by finding the
solution x
to
and , respectively.

I1(.) is the modified Bessel
function of order 1.

Example

The following example shows how to use the Power Law model
in RGA. Table 1 shows the failure times for each
unit in a sample of 11 systems in a fleet. The end time is the last recorded known
age when the analysis was performed. The end time for each unit is less than the
last failure time (if the unit has failed), thus the data set is time terminated
data.

Step 2: Add the failure data to the data sheet, then click the
Calculate icon to estimate the parameters using
the MLE method with Crow
confidence bounds. As shown below, the estimated parameters
are ,
and .

Step 3: Click the QCP icon to open the Quick Calculation Pad.
Select the Reliability option and enter the inputs as shown below. Using
a Time/Stage value of 2000 and a Mission Time value of 200, and
using two-sided confidence bounds with a confidence level of 0.9, the
conditional reliability is 0.8577 with an upper confidence bound of 0.9389 and a
lower confidence bound of 0.7322.

Conclusion

This article briefly introduced the Power Law model, and then described the
equations for estimating the conditional reliability and its bounds. Finally an
example using RGA provided an illustration of using the Power Law
model.